Questions on (ordinary) differential equations. For questions specifically concerning partial differential equations, use the (pde) tag.

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137 views

A question regarding Frobenious method in ODE

Suppose $b(x),c(x)$ are real functions analytic at 0. Let $b(x)=\sum_{i=0}^\infty b_ix^i, c(x)=\sum_{i=0}^\infty c_ix^i$ on $(-R,R)$. Suppose $r$ is a double root of $r(r-1)+b_0r+c_0=0$. It is well ...
4
votes
0answers
66 views

Solving second order nonlinear ODE

Having the following second order ordinary differential equation: $$ \ddot{x} = a \cos(x) $$ where, $a$ is a constant. What's an approach to solve this kind of equation?
4
votes
1answer
117 views

Asymptotic estimate of an oscillatory differential equation

Let $f\in C^1(\mathbb{R} ,\mathbb{R} )$ and satisfying the condition: $$ \forall x >0, \quad f(x)>0, \forall x<0 , \quad f(x)<0 $$ Let $(\alpha, \beta) \in \mathbb{R^2}$. ...
4
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0answers
226 views

Modelling a Water Rocket. Requires Some Validation and Help. ( WARNING : Extremely Long but Interesting Post )

Good day people of math.stackexchange.com This is a pet project that I plan to use to convince my Prof that I would rather try something similar to this than to do the prescribed project. Edit : ...
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1answer
1k views

Solving an initial value ODE problem using fourier transform

I am a physics undergrad and studying some transform methods. The question is as follows: $y^{\prime \prime} - 2 y^{\prime}+y=\cos{x}\,\,\,\,y(0)=y^{\prime}(0)=0\,\,\, x>0$ I am having some ...
4
votes
1answer
139 views

Is Euler's lemma of fluid mechanics a nonlinear version of Liouville's theorem of ODEs?

Liouville's Theorem Consider the following linear system of ordinary differential equations: $$\tag{1} \dot{\mathbf{x}}=A(t)\mathbf{x}(t).$$ Let $\mathbf{x}_1, \mathbf{x}_2, \ldots, ...
4
votes
2answers
110 views

Time required to reach the goal when an object will be slowing down incrementally based on distance travelled?

I was thinking about this when flying on the plane which was approaching and slowing down. Assume an object is approaching its target which is at a certain initial distance d at time t0. It starts ...
4
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1answer
836 views

Convert Airy's equation, $y'' - xy = 0$, into Bessel equation, $t^2u'' + tu' + (t^2 - c^2)u = 0$

My professor has said that this will be an easy homework exercise. He suggested using change of variable $t = \dfrac{2}{3}x^{3/2}$, and then removing the first derivative term of the form $p(t) \dfrac ...
3
votes
1answer
57 views

Analysis of stability of a linearized ODE with a periodic solution

I am asked to find the stability of the following ODE: \begin{equation*} \dot{y} = y^{2} + 2\cos(t)\sin(t) - \sin^{4}(t) \end{equation*} by linearizing around a particular solution $\eta = ...
3
votes
3answers
417 views

Fourier Series for $|\cos(x)|$

I'm having trouble figuring out the Fourier series of $|\cos(x)|$ from $-\pi$ to $\pi$. I understand its an even function, so all the $b_n$s are $0$ $$a_0 = \frac 2 \pi \int_0^\pi |\cos(x)|\,dx = ...
3
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2answers
61 views

Finding the general solution to a system of differential equations

How can I solve the following system of differential equations? I am getting confused with the constants of integration... $$\dot{x}=2x-(2+y)e^{y}$$ $$\dot{y}=-y$$ I know that $y=Ce^{-t}$ and the ...
3
votes
2answers
101 views

Liapunov Function for a Differential Equation

I have been trying to find a Liapunov function which would give me information about the stability of the following system of differential equations, however, I am not able to come up with any. The ...
3
votes
1answer
164 views

Small question about ODE

i have this question : Given three parameters $L,a$ et $\alpha$, we consider the differential equation : $$(E)\qquad x''+\alpha x' +a x + \sin x =L, \ > t\geq0$$ 1) Show that the ...
3
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3answers
413 views

Power Series Solution for $e^xy''+xy=0$

$$e^xy''+xy=0$$ How do I find the power series solution to this equation, or rather, how should I go about dealing with the $e^x$? Thanks!
3
votes
2answers
516 views

First-order nonlinear ordinary differential equation

How to solve this differential equation: $$x\frac{dy}{dx} = y + x\frac{e^x}{e^y}?$$ I tried to rearrange the equation to the form $f\left(\frac{y}{x}\right)$ but I couldn't thus I couldn't use $v = ...
3
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1answer
171 views

differential system on the torus

In a recent topic I've studied on complex analysis I had to study the differential system on the torus $\mathbb T^2:$ $$\begin{cases}\frac{\partial}{\partial y}u-\frac{\partial}{\partial ...
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1answer
369 views

Explain the error term in Euler method

Task: I had to find out some estimates for M and L to make sure the proportional accucrazy is not above $10^{-4}$ in the Euler method with the problem below. I am trying to understand the page 672 on ...
3
votes
2answers
244 views

What is the formal definition of $d$, or $\partial$, in differation and integration

This might sound a bit like a silly question, but i'm a second year math student, and so far i've encountered $d$ or $\partial$ in many cases ofcourse (mostly in calculus :)). Those letters or symbols ...
2
votes
1answer
52 views

Under what conditions can a function $ y: \mathbb{R} \to \mathbb{R} $ be expressed as $ z z' $?

This is a follow-up to Under what conditions can a function $ y: \mathbb{R} \to \mathbb{R} $ be expressed as $ \dfrac{z'}{z} $?. It turns out that in that case, \begin{align} \text{$ y = ...
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0answers
56 views

modified ODE has same trajectories as original system and associated flow is defined for all $t \in \mathrm{R}$ [closed]

I really don't know where to start with this problem. Consider the differential equation $\dot{x} = f(x)$ with $f \in C^1(\mathrm{R}^n,\mathrm{R}^n)$. Consider the following modified differential ...
2
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2answers
131 views

Finding the Asymptotic Curves of a Given Surface

I have to find the asymptotic curves of the surface given by $$z = a \left( \frac{x}{y} + \frac{y}{x} \right),$$ for constant $a \neq 0$. I guess that what was meant by that statement is that surface ...
2
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1answer
1k views

Polar coordinates differential equation

I have the following ODE: $$\dot x=-y(x^2+y^2), \dot y=x(x^2+y^2)$$ I want to sketch the phase portrait (manually) and I want to find the flow $\phi_t$, the orbit $O(x_0)$ and the limit set ...
2
votes
2answers
97 views

On the existence of a particular solution for an ODE

The problem asks to find a bounded $u(\cdot) \in \mathcal{C}^2(\mathbb{R})$ such that $$u''+u'-2u=f$$ where $f$ is a bounded continuous function on the real line. [Observations, Editted] We can ...
2
votes
1answer
301 views

Stability of nonlinear system with borderline linearization

I have the following nonlinear system: \begin{align} ...
2
votes
3answers
114 views

Getting equation from differential equations

I have: $\dfrac {dx} {dt}$=$-x+y$ $\dfrac {dy}{dt}$=$-x-y$ and I am trying to find $x(t)$ and $y(t)$ given that $x(0)=0$ and $y(0)=1$ I know to do this I need to decouple the equations so that I ...
2
votes
1answer
107 views

Solving $f_n=\exp(f_{n-1})$ : Where is my mistake?

I was trying to solve the recurrence $f_n=\exp(f_{n-1})$. My logic was this : $f_n -f_{n-1}=\exp(f_{n-1})-f_{n-1}$. The associated differential equation would then be $\dfrac{dg}{dn}=e^g-g$. if ...
2
votes
1answer
650 views

Possible ways to do stability analysis of non-linear, three-dimensional Differential Equations

For example Lorenz system, $$ \frac{d}{dt}\begin{pmatrix} x\\ y\\ z \end{pmatrix}=\begin{pmatrix} -\sigma & \sigma & 0\\ \rho & -1 & -x\\ y & 0 & -\beta ...
2
votes
1answer
494 views

Show that Bessel function $J_n(x)$ satisfies Bessel's differential equation.

here is the question: For each positive integer $n$, the Bessel function $J_n(x)$ may be defined by $$J_n(x) = \frac{x^n}{1\cdot 3\cdot 5\cdots(2n-1)\pi}\int^1_{-1}(1-t^2)^{n-1/2}\cos(xt) \, dt$$ ...
2
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2answers
465 views

Clarification of Frobenius method roots

The frobenius method states that for repeated roots or roots that differ by an integer, an alternative method must be used to find the second solution once one is found. When they say "roots that ...
2
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2answers
823 views

second derivative of the inverse function

I know that the derivative of the inverse function of $f(x)$ is $g'(y) = \frac{1}{f'(x)}$ But how to derive the formula for the second derivative of g(y) knowing that $\left[\frac{1}{f(x)}\right]' = ...
2
votes
2answers
157 views

Inhomogeneous equation

Let $A$ be an $n\times n$ matrix and $\beta$ a constant. Consider the special inhomogeneous equation $$\dot x = Ax + p(t)e^{\beta t},$$ where $p(t)$ is a vector all whose entries are polynomials. Set ...
2
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1answer
97 views

The system $x'=Ax$ is an attractor if and only if there is a positive quadratic form q such that $Dq(x)\cdot A(x)<0$ for all x

I need to show this result: Given the system of ODEs $x'=Ax$, the origin, $0$, is an attractor (equivalently, all the eigenvalues of the real matrix $A$ are negative) if and only if there exists a ...
2
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1answer
223 views

To find the closed form of $ f^{-1}(x)$ if $3f(x)=e^{x}+e^{\alpha x}+e^{\alpha^2 x}$

$$3f(x)=e^{x}+e^{\alpha x}+e^{\alpha^2 x}$$ where $\alpha=e^{\frac{2\pi i}{3} }$ I would like to find a closed form of $ f^{-1}(x)$ $$f(x)=\sum \limits_{k=0}^\infty \frac{x^{3k}}{(3k)!}$$ We can ...
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2answers
1k views

Use of Legendre's equation.

For some weeks have been studying Legendre polynomial as a solution to this equation. $$ (1-x^2)\frac{d^2}{dx^2}f(x)-2x\frac{d}{dx}f(x)+n(n+1)f(x)=0.$$ I've found them very interesting to learn from ...
2
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2answers
543 views

Help on differential equation $y''-2\sin y'+3y=\cos x$

$y''-2\sin y'+3y=\cos x$ I'm trying to solve it by power series, but I just can't find the way to get $\sin y'$. Is there any special way to find it?
2
votes
1answer
369 views

Existence of global solution of Riccati equation

Consider a Riccati differential equation $$ \dot P + A(t)^{T}P + PA(t) -PB(t)R(t)B(t)^{T}P + Q(t) = 0,\;\;\; P(t_0) = P_0 = P_0^{T} \geqslant 0 $$ where $Q(t) = Q(t)^{T} \geqslant 0$, $R(t) = ...
2
votes
4answers
987 views

$y'''-y=x^{2}$ has solution — `“multiplicity”`?

The page 667 of the book (sorry not in English) claims $y'''-y=x^{2}$ to have the solution $$y(x)=C_{1}e^{x}+e^{-x/2}\left(C_{2} \cos \left( \frac{\sqrt{3}x}{2} \right)+C_{3} ...
2
votes
3answers
112 views

Some double angle identity to solve $(2x^{2}+y^{2})\frac{dy}{dx}=2xy$?

For some reason, I cannot see a clever way to solve this (I know the way of doing it like in Wolframalapha) but I am pretty sure there is a double angle identity to crack this puzzle. Could someone ...
2
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3answers
147 views

The equation $(x-2xy-y^2)\frac{dy}{dx}+y^2=0$

Can one give a hint how to solve the following equation? $(x-2xy-y^2)\frac{dy}{dx}+y^2=0$ Thanks in advance.
2
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2answers
475 views

Poincare-Bendixson Theorem

Can someone sketch some ideas of how to use the Poincare-Bendixson Theorem to prove that there must be a fixed point contained inside a periodic orbit?
1
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4answers
99 views

Separable First Order Ordinary Differential Equation with Natural Logarithms

Please check my work: $$xy' = 5y$$ $$\int\frac{dy}{y} = 5\int\frac{dx}{x}$$ $$\ln y = 5\ln x + c$$ $$y = 5x + c$$ Is this correct?
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1answer
52 views

Question about Poisson formula

We have the Laplace equation in polar coordinates: $$u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{\theta \theta}=0, 0 \leq r <a, 0 \leq \theta \leq 2 \pi$$ With the separation of variables, the solution ...
1
vote
1answer
42 views

Second order differential equation with non constant coefficients

I want to solve the following differential equation: $$ 2f'(x)(2x+1)+\frac{\kappa}{2}f"(x)x(x+1)=f(x)(\frac{-2b}{x+1}+\frac{2c}{x}+2a) $$ where $\kappa,a,b,c$ are arbitrary positive constants. Is ...
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1answer
39 views

Let $y'' + p(x)y' + q(x)y = 0$ , where $p(x)$ and $q(x)$ are continuous. Prove that the zeroes of $y$ are isolated.

Let $p$ and $q$ be continuous, and let $y$ be any solution of $y′′(x) + p(x)y′(x) + q(x)y(x) = 0$ that is not identically zero. Then zeroes of $y$ are isolated, in the precise sense that for any ...
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1answer
63 views

Interpreting another proposition full of symbols

Could someone help me interpret the following proposition full of symbols? I've been struggling to comprehend it. Thanks in advance. Proposition: Suppose that $f:\mathbb{R^n} \rightarrow ...
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1answer
72 views

Questions concerning the differential operator

Consider the differential equation:- $a \phi + (bD^3 - cD)w =0$, where $a, b$ and $c$ are constants, $D$ denotes the differential operator $\dfrac{d}{dx}$, and $w$ is a function of $x$. I'm ...
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1answer
86 views

Question about an O.D.E

I have this theorem: Suppose that $U$ is a neighborhood of $\theta$ in a Hilbert space $H$ and that $f\in C^2(U,\mathbb{R}^1)$. Assume that $\theta$ is the only critical point of $f$ and that ...
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0answers
62 views

Inverse Laplace Transform using Jordan's Lemma?

Following is the question that i am trying to solve: "Consider a second order linear ODE $x\dfrac{d^2y}{dx^2}+x\dfrac{dy}{dx}+(3-2x)y=0$ A) Find the solution employing Laplace integrals by ...
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1answer
73 views

Counter-example to Cauchy-Peano-Arzela theorem

I was looking for a counter-example to Cauchy-Peano-Arzela theorem. I found this paper (in french) from Dieudonné. [acta.fyx.hu] Take $E = c_0$ to be the space of real sequences converging to $0$, ...
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0answers
23 views

Getting Eigenvalues Into a Differential Operator?

Following Butkov, a second order ode $$A(x)y'' + B(x)y' + C(x)y = D(x)$$ can always be brought into Sturm-Liouville form $$\tfrac{d}{dx}[p(x)y'] - s(x)y = f(x)$$ after multiplying across by ...